5 5 is equal to 4x7728; but this fraction must be taken 3 times to form the first fraction; hence of =3x1 of =3x =l, a result which is obtained by multiplying together the numerators and denominators of the compound fraction. When the compound fraction consists of more than two simple ones, two of them can be reduced to a simple fraction as above, and then this fraction may be reduced with the next, and so on. We therefore have the following RULE. Ans. . I. Reduce all mixed numbers to their equivalent improper fractions by Case II. II. Then multiply all the numerators together for a numerator and all the denominators together for a denominator : their products will form the fraction sought. Ex. 2. Reduce { of of to a simple fraction. Here 1 ***=: 3. Reduce of j of 4 to a simple fraction. Here X3x4=26=1=4 by dividing the numerator and denominator first by 9 and then by 2, as shown in Case III. Or, š xx=4, by cancelling the 3's and 6's in the numerator and denominator. By cancelling or striking out the 3's we only divide the numerator and denominator of the fraction by 3; and in cancelling the B's we divide by 6. Hence the value of the fraction is not affected? striking out like figures, which should alwa done when they multiply the numerator an minator. 4. Reducer of of is to a simple fraction. Here 4x}xt=ws==: Ans. Or, *xx=1= Ans. Ans. 42=1027 Ans. H=27 CASE VI. Ø 130. To reduce fractions of different denomina. tors to equivalent fractions having a common denominator. RULE. I. Reduce compound fractions to simple ones, and whole or mited numbers to improper fractions. II. Then multiply each one of the numerators hy all the denominators except its own, for the new numerators, and multiply all the denominators together for a common denominator : the common denominalor placed under each of the new numerators will form the several fractions sought. Ex. 1. Reduce }, }, and f to a common denomi. nator. 1x3x5=15 the new numerator for 4*3*2=24 do do Therefore, jf. ff and ft, are the equivalent frac. tions. It is plain, that this reduction does not alter the values of the several fractions, since the numerator nd denominator of each are multiplied by the same . mber. $ 118. When the numbers are small the whole work may be performed mentally. Thus, = 16 14 16 Here we find the first numerator by multiplying 1 by 4 and 5; the second, by multiplying 1 by 2 and 5; the third by multiplying 2 by 4 and 2; and the common denominator by multiplying 2, 4 and 5 together. Ex. 3. Reduce 2}, and į of ļ to a common denominator. 2}=1; and of t=t. 1, tr, and; the answers. 4. Reduce 51, $ of , and 4, to a common denominator. Ans. 11, it, and if 5. Reduce 1, 235, and 37, to a common denominator. Ans. 184, 18, and 2764 6. Reduce 4, 3, 2, to a common denominator. Ans. 40, 42, and 1444. Note. § 131. It is often convenient to reduce fractions to a common denominator by multiplying the numerator and denominator of each fraction by such a number as shall make the denominators the same in both. 7. Let it be required to reduce į and f to a common denominator. We see at once that if we multiply the numerator and denominator of the first fraction by 3, and the numerator and denominator of the second by 2, that they will have a common denominator. The two fractions will be reduced to 3 and 4. 8. Reduce and to a common denominator: Here, fx3=is; and f*f=ts. Ans. 's and to QUESTIONS. $123. What is Reduction of Vulgar Fractions ? ở 124. When is a fraction said to be in its lowest terms ? $ 125. How do you reduce an improper fraction to its equivalent whole or mixed number? Ø 126. How do you reduce a mixed number to its equiva. lont fraction ? Will the fraction be proper, or iniproper ? § 127. How do you reduce a fraction to its lowest termina ? Will the value be altered by the reduction ? Why? $ 128. How do you reduce a whole number to a fraction having a given denominator ? $ 129. How do you reduce a compound fraction to a sim. ple one ? When you have the same multiplier in the nume. rator and denominator, what do you do? Does this altor the value of the fraction ? 130. How do you reduce fractions to a common denomi. nator? Does this reduction change the values of the seve. ral fractions ? Why not $ 131. What is a second method of reducing fractions to a common denominator ? REDUCTION OF DENOMINATE FRACTIONS. $ 132. We have seen $71, thal a denominate number is one in which the kind of unit is denomi. nated, or expressed. For the same reason, a denominate fraction is one which expresses the kind of unit that has been divided. Such unit is called the unit of the fraction. Thus of a £ is a denominate fraction. It expresses that one £ is the unit which has been divided, that this unit has been divided into 3 equal parts, and that 2 of these parts are taken in the fraction. The fraction of a shilling is also a denominate fraction, in which the unit that has been divided iss one shilling. These two fractions are of different denominations, the unit of the first being one pound, and that of the second, one shilling. Fractions are therefore of different denominations when they express parts of different units, and of the same denomination when they express parts of the same unit. $ 133. Reduction of denominate fractions consists in changing their denominations without altering their values. For example, to reduce of a £. to the denomination of shillings, or of a shilling to the denomination of pounds. CASE I. $ 134. To reduce a denominate fraction from a lower to a higher denomination. RULE. I. Consider how many units of the given denomination make one unit of the next higher, and plate 1 over that number forming a second fraction. II. Then consider how many units of the second denomination make one unit of the denomination next higher, and place 1 over that number forming a third fraction; and so on, to the denomination to which you would reduce. III. Connect all the fractions together, forming a compound fraction; then reduce the compound fraco tion to a simple one by Case V. Ex. 1. Reduce of a penny to the fraction of a £. Here of tg of zv=£no. The given fraction is of a penny. But one penny is equal to ty of shilling: hence } of a penny is equal to f of the of a shilling. But one shilling is equal to the of a |